‘Leading Conservatives are increasingly aware that many people are changing their minds on immigration.’ Home secretary Sajid Javid and prime minister Theresa May.
Photograph: Steve Parsons/PA

The campaign to secure a new referendum on remaining in the EU faces a relentlessly ticking clock. Converting Jeremy Corbyn’s frontbench and enough Conservative backbenchers to the cause would leave little time to hold a public vote by the time Britain is due to leave the EU. Unless negotiations collapse entirely, it seems a long shot.

The public are both less bothered by immigration and more positive about migrants who come here. Immigration is now a considerably lower priority for voters than the NHS, and its economic and cultural impacts are viewed more positively – though most would still prefer the pace of migration to slow down.

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Gove attributed the change in attitudes to a greater sense of control after the referendum vote. His critics suggested the shift might be despite Brexit, rather than because of it. These may be complementary rather than competing explanations: different people change their views for different reasons. When Ipsos Mori asked those with more positive attitudes why their views had shifted, equal proportions said that they were either more aware of positive contributions now, or were expecting Brexit to lead to change.

So this shift in public opinion may not be one that would sweep remain to victory in a new vote. Nine out of 10 voters say they would vote the same way on Brexit, despite the shift on immigration. A new referendum would reheat and repolarise the immigration debate, returning to what most people still see as a binary question: remain or leave, free movement or the ability to set limits. Unless EU governments made the sort of offer which David Cameron could not secure in 2016, pro-EU voices would again divide between defending free movement or campaigning to reform it (as half of remain voters would prefer) – or simply just trying to change the subject as soon as possible.

In the post-referendum landscape, we face a new, less binary question on immigration: what should we do now? Even two years on, this “what next?” debate has yet to begin in earnest, given the government’s tendency to delay substantive choices about Brexit.

Ministers should have more confidence in the public. The National Conversation on Immigration has seen British Future and Hope Not Hate hold citizens’ panels in 60 towns and cities across every nation and region to inform the home affairs select committee investigation into future immigration choices. It has found that most people are “balancers”: open to students and skills, preferring controls on low-skilled migration, but open to using them pragmatically where jobs need filling. Low trust in the competence of governments is combined with warmth towards migrants who come to contribute to our society.

There is no guarantee that attitudes may not shift back. However, the post-referendum trends do also reflect a gradual longer-term shift, particularly across generations, linked to the rise in higher education and increased social contact in a more diverse society – though social polarisation explains how more positive general attitudes can coexist with a spike in reported hate crime from a shrinking but vociferous racist fringe.

If a new referendum would likely replay the tense immigration debate, there is still an opportunity to build a consensus on a managed and welcoming post-Brexit system. There should be greater political accountability, with an annual Migration Day – similar to the budget – preceded by public engagement around the country that addresses both the pressures and gains of migration.

A new migration deal with Europe that is as open to European skills as now, with pragmatic controls on low-skilled migration, would command a broad public and political consensus in the UK. Nobody can know how the EU would respond until the UK has a negotiating position, but a constructive offer could pave the way towards a close post-Brexit partnership with the EU that most MPs across the Commons would back.

The new home secretary, Sajid Javid, has a big opportunity. Brexit and Windrush offer a reset moment for immigration – if he has the confidence to press the reset button, that is. The emerging Javid-Gove-Davidson axis offers a powerful alliance to break out of the referendum trenches inside government, and develop a position with cross-party appeal.

Yet government policy has been slow to change. Refusing visas to 1,500 doctors whom NHS trusts want to employ is the most egregious example of government policy being tougher than the British public viewpoint – as it has been on students, EU nationals and the treatment of the Windrush generation.

Leading Conservatives are increasingly aware that many people are changing their minds on immigration. Their problem will be persuading a stubborn prime minister to change hers too.

• Sunder Katwala is director of British Future and former general secretary of the Fabian Society